Set Theory: Solved Examples

Q.1. There are 300 students in a class 10th. Among them, 80 students are learning both History & Geography. A total of 180 students are learning History. If every student is learning at least one of these two subjects, how many students are learning Geography in total
A. 240
B. 200
C. 180
D. 150
Answer : Option B
Solution : Every student is learning at least one of these two subjects. Hence there is no one who fall in the category ‘neither’.
It is mentioned in the problem that a total of 180 are learning History.
Now, 180 are learning History and 80 are learning both the subjects. This means that 180 – 80 = 100 are learning only History.
So, out of a total of 300 students, 100 are learning only history which clearly means that rest 200 are learning Geography (whether only geography or with history)
Therefore, total number of students learning Geography = 200.
Q.2. In a survey conducted to know people’s preference for tea and coffee, 160 people preferred tea while 120 people preferred coffee. 40 of them liked both and may prefer any of the beverages. If there was no one in the group who didn’t prefer at least one of the beverages, then on how many people was this survey conducted?
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A. 80
B. 160
C. 120
D. 240
Answer: Option D
Solution: Let n(A) be the number of people who prefer tea , the number of people who prefer coffee be n(B) and n(A∩B) be the people who liked both.
Using the formula: n(A∪B) = n(A) + n(B) – n(A∩B)
n(A∪B) = 160 + 120 – 40 = 240
Q.3. A toy factory has three machines A, B & C and 240 workers. Each worker knows to operate at least one of the three machines. If there are 195 persons who know how to operate machine A, 180 who know how to operate machine B and 165 who know how to operate machine C, 120 know how to operate machine A & B , 125 know how to operate machine B & C , 130 know how to operate machine A & C . How many workers know how to operate all these three machines?
A. 75
B. 90
C. 120
D. 45
Answer: Option A
Solution: Let A be the set of people who know how to operate machine 1
Let B be the set of people who know how to operate machine 2
Let C be the set of people who know how to operate machine 3
A∪B∪C = A + B + C - (A n B + B n C + C n A) + (A n B n C)
240 = 195 + 180 + 165 – ( 120 + 125 + 130) + x
X = 75
Q.4. In a survey, it was found out that 100 people don’t use Laptop, Mobile or wrist watches. 80 persons use all the three gadgets. There are 150 who use Laptop and Mobile, 200 who use Mobile and Wristwatch and 200 who use Laptop and Wristwatch. The number of people who use only Laptop, only Mobile and only Wristwatch is equal. If this survey was conducted on 1000 persons ,How many people use only Wristwatch?
A. 200
B. 90
C. 150
D. 900
Answer: Option B
Solution: Let the number of people that use only Facebook = only Twitter = only Whatsapp = x
AUBUC = A + B + C - (A n B + B n C + C n A) + (A n B n C)
We can see that x + x + x + 80 + 200 + 200 + 150 + 100 = 1000
⇒ 3x = 1000 – 730
⇒ x = 270/3 = 90
Q.5. In a class of 120 students, 40 students passed in English, 60 students passed in Hindi, 40 students passed in Punjabi. Each student passed in at least one subject. 10 students passed in both English and Hindi, 12 students passed in both Hindi & Punjabi, 15 students passed in Punjabi & English . How many students passed in all the subjects?
A. 16
B. 17
C. 18
D. 20
Answer: Option B
Solution : Let A be the set of people who passed in English
Let B be the set of people who passed in Hindi
Let C be the set of people who passed in Punjabi
A∪B∪C = A + B + C - (A n B + B n C + C n A) + (A n B n C)
120 = 40 + 60 + 40 -10-12-15 + x
X = 120 – 103
X = 17
Q.6. In a party of 120 people, 60 people will choose Ice tea, 24 people will choose Ice cream and 17 people will choose Cold Coffee. 12 people chose Ice Tea & Ice cream, 8 people chose Ice Cream & Cold Coffee., 3 people chose Cold Coffee & Ice tea. 1 person consumed all three. How many opted for none of the three things mentioned?
A. 29
B. 21
C. 41
D. 27
Answer: Option C
Solution: A∪B∪C = A + B + C - (A n B + B n C + C n A) + (A n B n C)
A is the set of people who chose Ice Tea = 60
B is the set of those who chose Ice Cream = 24
C is the set of those who chose Cold Coffee = 17.
So, AUBUC = 60 + 24 + 17 - (12 + 8 + 3) + 1 = 79.
Number of people who selected none of the 3 items = 120 - 79 = 41.
Q.7. Of the 400 persons, who were interviewed for a post at an MNC, 200 owned a four-wheeler, 140 owned a debit card and 280 owned a wrist watch. 80 of them owned both, a four-wheeler and a debit card, 60 owned both, a debit card and a wrist watch and 120 owned both, a four-wheeler and wrist watch and 10 owned all three. How many candidates had none of the three objects mentioned?
A. 50
B. 30
C. 20
D. 38
Answer: Option C
Solution: Number of persons who owned none of the three = Total number of persons -number of persons who owned at least one of three devices.
The total number of persons = 400.
Number of candidates who owned at least 1 of the 3 objects = A ∪ B ∪ C, where A is the set of people who owned a four wheeler, B is the set of those who owned a debit card and C is the set of those who owned a wrist watch.
As A∪B∪C = A + B + C - {A n B + B n C + C n A} + A n B n C. So, A∪B∪C = 200 + 140 + 280 - {80 + 60 + 120} + 20
Or A∪B∪C = 380.
380 candidates who attended the interview had at least one of the three gadgets, so 400 - 380 = 20 candidates had none of three objects.
Q.8. The schedule of 300 second year students was examined. It was found that 180 opted for Quantitative Aptitude course, 150 opted for a Verbal course and 50 opted for both a Quantitative Course & Verbal course. How many students opted for neither Quantitative Aptitude Course nor Verbal course?
A. 20
B. 25
C. 30
D. 40
Answer: Option A
Solution: Let A be the set of people who opted for Quantitative Course
Let B be the set of people who opted for Verbal Course
AUB = A + B - (A n B)
x = 180 + 150 -50
x = 280
So, out of 300 students, 280 of them have opted for something or other. Therefore, 20 students have opted for none of the subjects.
Q.9. In a class of 80 students, 44 students enrolled to study Spanish & 24 students enrolled to study both French and Spanish. If all the students of the class enrolled for at least one of the two subjects mentioned, then find the total number of students enrolled for only French and not Spanish?
A. 60
B. 20
C. 36
D. 56
Answer: Option C
Solution: Let A be the set of students who enrolled for French Language and B be the set of students who enrolled for Spanish Language.
So, (A ∪ B) is the set of students who enrolled for at least one of the two languages. As the students of the class have enrolled for at least one of the two languages, so A ∪ B = 80
A ∪ B = A + B - (A n B)
i.e, 80 = A + 44 - 24
or A = 60 which is the set of students who enrolled for French and includes those who enrolled for both the languages.
But, we need to find out the number of students who enrolled for French only= Students enrolled for French - Students enrolled for both French & Spanish.
= 60 - 24 = 36
Q.10. In class 12th, 70% of the students study Accountancy and 40% of the students study Business Studies. If 15% of the total students study both Accountancy and Business Studies, what % of the students do not study either of the two subjects?
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A. 5%
B. 25%
C. 10%
D. 15%
Answer: Option A
Solution: (A ∪ B) = A + B - (A n B), where (A ∪ B) represents the set of students who opted for at least one of the two subjects- Accountancy or Business Studies and (A n B) represents the set of people who opted for both the subjects- Accountancy or Business Studies.
Note:
(A ∪ B) = A + B - (A n B) => (A ∪ B) = 40 + 70 - 15 = 95%
95% of the total students study at least one of the two subjects- Accountancy or Business Studies.
So, the remaining (100 - 95)% = 5% of the total students do not study either of the two subjects.